Unit 1 Test Review - Piecewise Functions

Unit 1 - Study Guide Quizizz

Topics you'll review in this Quizizz:

• Define Piecewise Functions

  • Evaluate Piecewise Functions - From a Graph + Equation

  • Determine the Domain & Range of Piecewise Functions

  • Connect Piecewise Functions to Real World Context

• Define Absolute Value Functions

  • Determine the Domain & Range of Absolute Value Functions

  • Determine the Transformations of Absolute Value Functions

    • Match the graph to the absolute value equation

• Define Step Functions

  • Evaluate Step Functions

  • Connect Step Functions to Real World Context

• Discontinuity & Limits

  • Identify a discontinuity type @ specific x-value

  • Determine one-sided limits, both-side limits given limit notation

Define Piecewise Functions

Evaluating Piecewise Defined Functions

• To EVALUATE a Function is to: Replace (substitute) its variable with a given number or expression

• To Evaluate a piecewise function, We have to determine which interval the input belongs to.

Multiple Choice

• Steps:

• - Identify which interval -1 is in.

• - Use the correct equation to evaluate.

• - Substitute the x value into the equation

• - Simplify, What is the answer?

• 0

• 1

• 2

• 3

Multiple Choice

What is f(3)??

• -5

• 4

• 6

• 7

Multiple Choice

What is f(x) when x equals 0?

• -5

• 0

• 1/2

• 1

Multiple Choice

What is f(x) when x equals 4?

• -5

• 0

• 3

• 6

Multiple Choice

What is f(x) when x equals 8?

• 0

• 5

• 11

• 16

Multiple Choice

A house painter charges $12 per hour for the first 40 hours he works, $18 for the ten hours after that, and $24 for all hours after that.

How much does he earn for a 70 hour week?

• $1020

• $1140

• $840

• He works for free!

Multiple Choice

?

• $12.75

• $35

• $43.40

• $27.75

Piecewise Functions

• Remember to look at the inequality symbols for open or closed circles and to see which way to connect the dots.

• You also need to look here for function notation.

• f(0) = -4 and f(2) = 1

Multiple Choice

Evaluate f(-5)

• 0

• 2

• 5

• -4

Define Absolute Value Functions

• Determine the Domain & Range of Absolute Value Functions

• Determine the Transformations of Absolute Value Functions

  • Match the graph to the absolute value equation

Domain and Range of a Function

The range of a function is the set of all possible output values (y)​ for the function.​ Includes all of the numbers on the vertical number line.

Range

The domain of a function is the set of all possible input values (x)​ for the function. Includes all the numbers on the horizontal number line

Domain

Multiple Choice

The domain of the function f(x)=x​ is the set of all real numbers.

The domain of the function f(x)=|x|​ is

• The set of real numbers > 0

• The set of real numbers ≥ 0

• The set of real numbers ≤ 0

• The set of ALL real numbers

Multiple Choice

State the domain and range of given Absolute Value Graph.

• Domain: All Real Numbers ; Range: y > -3

• Domain: All Real Numbers ; Range: y < -3

Multiple Choice

What is the range of the graph?

• $$y\ge1$$

• $$y\ge0$$

• $$y\ge-1$$

• $$y\ge-2$$

Multiple Choice

State the domain and range of given Absolute Value Graph.

• Domain: All Real Numbers ; Range: y > 3

• Domain: All Real Numbers ; Range: y < 3

Graphing Absolute Value Functions with Transformations

PARENT FUNCTION

The parent function of the absolute value function is y=|x|.

The vertex is at (0,0) and opens upward.

Remember, you can change the parameters of the absolute value function to change the location of the vertex and the "openness" of the graph.

• ​a Compresses or stretches the graph vertically (Graph will Push TO and Pull AWAY from the X-axis)

• h Translates the graph of the absolute value function either to the left (+h) or right (-h) !

• k Translates the graph of the absolute value function either to the up (+k) or down (-k) !

Parameters of the Absolute Value Function - a, h, k:

The Quiz will have some questions on transformations. Make sure you have the parameter effects written in your notes!

Multiple Select

Select all that apply. (May have more than one correct answer)

f(x) = |x - 6| + 2

• Horizontal translation right 6 units

• Horizontal translation left 6 units

• Vertical translation up 2 units

• Vertical translation down 2 units

Multiple Choice

Which function defines the graph?

• f(x) = |x - 4| + 3

• f(x) = |x - 3| + 4

• f(x) = |-x + 4| + 3

• f(x) = |x + 4| + 3

Multiple Select

What is the equation of the graph below?

• y = - | x + 2| + 4

• y = | x - 2| - 4

• y = - | x - 2| + 4

• y = | x - 2 | + 4

Multiple Choice

Which function is graphed?

• f(x) = 2|x + 2| -4
• f(x) = 2|x - 4| + 2
• f(x) = |x + 2| - 4
• f(x) = -2|x - 4|

Multiple Choice

Choose the correct equation for the graph.

• $$y=\left|3x\right|$$

• $$y=\left|\frac{1}{3}x\right|$$

• $$y=-\left|\frac{1}{3}x\right|$$

• $$y=\left|x-\frac{1}{3}\right|$$

Multiple Choice

Choose the correct equation for the graph.

•  $$y=\left|x+2\right|$$

• $$y=\left|x-2\right|$$

• $$y=\left|x\right|+2$$

• $$y=\left|x\right|-2$$

Drag and Drop

If "a" is negative, the v in the graph will _____?

Placeholder

​

Drag these tiles and drop them in the correct blank above

reflect

shift up

shift left

shift right

shift down

get thinner (stretch)

get wider (Shrink)

Dropdown

In what direction will the graph translate if parameter "h" is positive?​

Select answer

Multiple Choice

Parameter "k" tells us how the graph will translate Vertically.

If "k" is negative, in what direction will the graph translate ?

• Left

• Right

• Up

• Down

Multiple Choice

Which of the graphs given is

y=|x+1|−4 ?

(Click on the picture to enlarge it so you can see)

• A

• B

• C

• y=D

Replace this with your body text.

​Duplicate this text as many times as you would like.

​

Step Functions

​

​

Multiple Choice

Q:

Which of functions represents the graph?

HINT- Match the graph to its equation. Look at the y value of the line segments and the notation of the domains *Remember open circle/closed circle notation

Multiple Choice

Evaluate  $$f\left(4\right)$$  

Hint: Given (x), Find y.

• 2

• 4

• 6

• 0

• DNE

Multiple Choice

Q: Tre can start wrestling at age 5 in Division 1. He remains in that division until his next odd birthday when he is required to move up to the next division level.

Which graph correctly represents this information?

Multiple Choice

Q: How much does it cost to rent the Karaoke machine for 3 days?

Hint: Evaluate with context. What is the input value (x) from the question? Find the corresponding output (y) from the graph.

• $125

• $100

• $150

• $75

Continuous

• The function at x< 0 has the same output value as the function at x > 0

Discontinuous

• The function defined for x = 2 is not the same as the function defined at x<2 or x>2
  

Multiple Select

What are the three types of discontinuities (shown in the image?

• Hole (removable)

• Vertical Asymptote (non removable)

• Jump Discontinuity ( non removable)

• transitive discontinuity (removable)

• reflective discontinuity

Multiple Choice

Is this piecewise function continuous?

• Yes

• No

Multiple Choice

Where is this piecewise function discontinuous?

• x = -2

• x = 2

• x = -1

• x = 1

Multiple Choice

Which Graph shows Continuity?

• Graph 1

• Graph 2

Limits and continuity

By Du Tran

​To understand what limits are, let's look at an example.

We start with the function f(x)=x+2

​

​

The limit of f at x=3 is the value f approaches as we get closer and closer to x=3. Graphically, this is the y-value we approach when we look at the graph of f and get closer and closer to the point on the graph where x=3.​

​For example, if we start at the point (1,3) and move on the graph until we get really close to x=3 then our y-value (i.e. the function's value) gets really close to 5.

​Similarly, if we start at (5,7) and move to the left until we get really close to x=3 y-value again will be really close to 5.

​For these reasons we say that the limit of f at x=3 is 5.

​

​You might be asking yourselves what's the difference between the limit of f at x=3 and the value of f at x=3, i.e. f(3).

​

​So yes, the limit of f(x)=x+2 at x=3 is equal to f(3), but this isn't always the case. To understand this, let's look at function g. This function is the same as f in every way except that it's undefined at x=3.

​Just like f, the limit of g at x=3 is 5. That's because we can still get very very close to x=3 and the function's values will get very very close to 5.

​

So the limit of ggg at x=3 is equal to 5, but the value of g at x=3 is undefined! They are not the same!

​

That's the beauty of limits: they don't depend on the actual value of the function at the limit. They describe how the function behaves when it gets close to the limit.

​

​We also have a special notation to talk about limits. This is how we would write the limit of f as x approaches 3

The symbol "lim" means we're taking a limit of something.

The expression to the right of "lim" is the expression we're taking the limit of. In our case, that's the function f.

The expression x→3 that comes below "Iim" means that we take the limit of f as values of x approach 3.

Multiple Choice

Which expression represents the limit of $$x^2$$  as x approaches 5?

• $$\lim\ 5^2$$  

• $$\lim_{x^2\to\ 5}$$  

• $$\lim_{x\to5}x^2$$  

• $$\lim_{x\to25}x$$  

A limit must be the same from both sides.

​

Now take, for example, function hhh. The y-value we approach as the x-values approach x=3 depends on whether we do this from the left or from the right.

When we approach x=3 from the left, the function approaches 4. When we approach x=3 from the right, the function approaches 6.

​

When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.

​

Multiple Select

Which of the limit exist?

• $$\lim_{x\to3}g\left(x\right)$$  

• $$\lim_{x\to5}g\left(x\right)$$  

• $$\lim_{x\to6}g\left(x\right)$$  

• $$\lim_{x\to7}g\left(x\right)$$  

Now take, for example, function hhh. The y-value we approach as the x-values approach x=3 depends on whether we do this from the left or from the right.

Multiple Choice

What is the limit as x --> 2 (This is a both sided limit)?

• 3
• 1
• Nonexistent
• Infinity

Multiple Choice

?

• 4

• -2

• 0

• D.N.E

Multiple Choice

?

• 0

• 1

• 2

• D.N.E

Multiple Choice

$$f\left(-5\right)\ =\$$  ____

• 3

• 1

• undefined

• DNE

Multiple Choice

$$\lim_{x\ \rightarrow\ -5\ }f\left(x\right)\ =$$  ____

• 3

• 1

• DNE

• undefined

• -3.5

Multiple Choice

$$\lim_{x\ \rightarrow\ 8\ }f\left(x\right)\ =$$  ____

• ∞

• 4

• DNE

• -1

Multiple Choice

$$\lim_{x\ \rightarrow\ 6}f\left(x\right)\ =$$  ____

• 0

• -5

• DNE

• 6

Multiple Choice

$$\lim_{x\rightarrow2^+\ }f\left(x\right)\ =$$  ____

• 8

• 2

• DNE

• -3

Multiple Choice

$$\lim_{x\rightarrow1}f\left(x\right)\ =$$  ____

• -1

• 2

• DNE

• -5